A conveyor belt is moving grain into a bin that is 3.20 m below the top of the conveyor belt. The grain does not slip on the conveyor belt that is inclined at 15.0° and they move at a constant speed of 6.00 m/s. In order for the conveyor belt to get the grain into the bin, what must the horizontal distance between the end of the conveyor belt and the bin be?
ah yes...the mastery module
so.....
are you gonna answer it?
To solve this problem, we can use the concept of projectile motion. The grain is projected horizontally from the end of the conveyor belt and falls vertically due to the force of gravity. We need to find the horizontal distance traveled by the grain before it reaches the bin.
First, let's break down the given information:
Initial velocity in the horizontal direction (u) = 6.00 m/s (constant speed)
Vertical distance (h) = 3.20 m
Vertical angle (θ) = 15.0° (angle of inclination)
To find the horizontal distance (x), we can use the equation of motion in the vertical direction:
h = ut sin(θ) - (1/2)gt²
Where:
u = initial velocity in the vertical direction (0 m/s since the grain is not projected vertically)
g = acceleration due to gravity (9.8 m/s²)
t = time taken to fall vertically to the bin
By rearranging the equation, we can solve for t:
t² = (2h) / g
Now, let's substitute the given values and calculate t:
t² = (2 * 3.20 m) / 9.8 m/s²
t² = 0.653 m
t ≈ 0.808 s
Next, we can find the horizontal distance (x) traveled by the grain using the equation:
x = ut cos(θ)
Substituting the known values:
x = (6.00 m/s) * cos(15.0°)
x ≈ 5.78 m
Therefore, in order for the conveyor belt to get the grain into the bin, the horizontal distance between the end of the conveyor belt and the bin should be approximately 5.78 meters.